For integrated circuits (e.g., VLSI chips) to work properly, the signals traveling along their gates and interconnects must be properly timed, and several factors are known to cause timing variations. As examples, variations in manufacturing process parameters (such as variations in interconnect diameter, gate quality, etc.) can cause timing parameters to deviate from their designed value. In low-power applications, lower supply voltages can cause increased susceptibility to noise and increased timing variations. Densely integrated elements and non-ideal on-chip power dissipation can cause “hot spots” on a chip, which can also cause excessive timing variations.
A classical approach to timing analysis is to analyze each signal path in a circuit and determine the worst case timing. However, this approach produces timing predictions that are often too pessimistic and grossly conservative. As a result, statistical timing analysis (STA)—which characterizes timing delays as statistical random variables—is often used to obtain more realistic timing predictions. By modeling each individual delay as a random variable, the accumulated delays over each path of the circuit will be represented by a statistical distribution. As a result, circuit designers can design and optimize chips in accordance with acceptable likelihoods rather than worst-case scenarios.
In STA, a circuit is modeled by a directed acyclic graph (DAG) known as a timing graph wherein each delay source—either a logic gate or an interconnect—is represented as a node. Each node connects to other nodes through input and output edges. Nodes and edges are referred to as delay elements. Each node has a node delay, that is, a delay incurred in the corresponding logic gates or interconnect segments. Similarly, each edge has an edge delay, a term of signal arrival time which represents the cumulative timing delays up to and including the node that feeds into the edge. Each edge delay has a path history: the set of node delays through which a signal travels before arriving at this edge. Each delay element is then modeled as a random variable, which is characterized by its probability density function (pdf) and cumulative distribution function (cdf). The purpose of STA is then to estimate the edge delay distribution at the output(s) of a circuit based on (known or assumed) internal node delay distributions.
The three primary approaches to STA are Monte Carlo simulation, path-based STA, and block-based STA. As its name implies, Monte Carlo simulation mechanically computes the statistical distribution of edge delays by analyzing all (or most) possible scenarios for the internal node delays. While this will generally yield an accurate timing distribution, it is computationally extremely time-consuming, and is therefore often impractical to use.
Path-based STA attempts to identify some subset of paths (i.e., series of nodes and edges) whose time constraints are statistically critical. Unfortunately, path-based STA has a computational complexity that grows exponentially with the circuit size, and thus it too is difficult to practically apply to many modern circuits.
Block-based STA, which has largely been developed owing to the shortcomings of Monte Carlo and path-based STA, uses progressive computation: statistical timing analysis is performed block by block in the forward direction in the circuit timing graph without looking back at the path history, by use of only an ADD operation and a MAX operation:
ADD: When an input edge delay X propagates through a node delay Y, the output edge delay will be Z=X+Y.
MAX: When two edge delays X and Y merge in a node, a new edge delay Z=MAX(X,Y) will be formulated before the node delay is added.
Note that the MAX operation can also be modeled as a MIN operation, since MIN(X,Y)=−MAX(−X,−Y). Thus, while a MIN operation can also be relevant in STA analysis, it is often simpler to use only one of the MAX and MIN operators. For sake of simplicity, throughout this document, the MAX operator will be used, with the understanding that the same results can be adapted to the MIN operator.
With the two operators ADD and MAX, the computational complexity of block based STA grows linearly (rather than exponentially) with respect to the circuit size, which generally results in manageable computations. The computations are further accelerated by assuming that all timing variables in a circuits follow the Gaussian (normal) distribution: since a linear combination of normally distributed variables is also normally distributed, the correlation relations between the delays along a circuit path are efficiently preserved.
However, it is common for high-end VLSI circuits to have level-sensitive latches and feedback loops—but most of the existing STA methods are not readily adaptable to accommodate analysis of circuits including these elements. When feedback loops are present, the latches—which are otherwise “transparent” in a timing sense (i.e., they do not affect timing)—may cause random timing variables to be self-dependent, in that the values of these variables in one iteration/cycle are dependent on their values in the prior iteration/cycle. STA methods for latch-based circuits have been proposed (see, e.g., M. C.-T. Chao, L.-C. Wang, K.-T. Cheng, and S. Kundu, “Static statistical timing analysis for latch-based pipeline designs,” IEEE/ACM International Conference on Computer Aided Design, 2004. ICCAD-2004, pp. 468-472, November 2004), but these generally do not address the issue of self-dependence. Those that do address self-dependence generally bear disadvantages which make them computationally expensive; for example, in R. Chen and H. Zhou, “Clock schedule verification under process variations,” IEEE/ACM International Conference on Computer Aided Design, ICCAD-2004, pp. 619-625, November 2004, graph sorting algorithms are proposed for dealing with feedback loops, but the computation complexity of these algorithms can grow exponentially with circuit size, thereby subjecting this methodology to many of the same disadvantages as for path-based STA.
Given that the trend in circuit fabrication is toward increased complexity with higher speed and lower size, there is clearly a pressing need for accurate methods of statistical timing analysis which compensate for issues raised by latches and feedback loops, and which are computationally efficient so that rapid design and testing is feasible.